Question

ential eqns common base (level 2)
tion
ve for x:
$625^{-2x + 9} = \left(\frac{1}{125}\
ight)^{-5x - 13}$

Explanation:

Step1: Rewrite bases as powers of 5

$625 = 5^4$, $\frac{1}{125}=5^{-3}$
Substitute into the equation:
$$5^{4(-2x+9)} = 5^{-3(-5x-13)}$$

Step2: Simplify exponents

Calculate each exponent:
Left side: $4(-2x+9) = -8x + 36$
Right side: $-3(-5x-13) = 15x + 39$
Equation becomes:
$$5^{-8x+36} = 5^{15x+39}$$

Step3: Set exponents equal

Since bases are equal, exponents are equal:
$-8x + 36 = 15x + 39$

Step4: Solve for x

Rearrange terms to isolate x:
$36 - 39 = 15x + 8x$
$-3 = 23x$
$x = \frac{-3}{23}$

Answer:

$x = -\frac{3}{23}$